Question

given n || m and < 1 ≅ < 2. prove p || r. n is parallel to m, < 1 is congruent to < 2. < 1 is congruent to < 3. < 2 is congruent to < 3. p is parallel to r. given, reflexive property, corresponding angles are congruent, converse of corresponding angles, symmetric property, alternate interior angles are congruent, transitive/substitution property, converse of alternate exterior angles, alternate exterior angles are congruent, converse of alternate interior angles

Explanation:

Step1: State the given information

Given \(n\parallel m\) and \(\angle1\cong\angle2\)

Step2: Use the property of parallel lines

Since \(n\parallel m\), \(\angle1\cong\angle3\) (Corresponding - angles are congruent when \(n\parallel m\))

Step3: Substitute using the given congruence

Given \(\angle1\cong\angle2\), and \(\angle1\cong\angle3\), so \(\angle2\cong\angle3\) (Transitive/Substitution property)

Step4: Apply the converse theorem

Since \(\angle2\cong\angle3\), then \(p\parallel r\) (Converse of corresponding - angles)

Answer:

The proof that \(p\parallel r\) is completed by first using the corresponding - angles property for \(n\parallel m\), then the transitive/substitution property with the given \(\angle1\cong\angle2\), and finally the converse of the corresponding - angles theorem.